Dr. Abdullah M. Elsayed

Department of Electrical Engineering

Damietta University – Egypt

010 60 79 1554

am.elsherif@du.edu.eg

Course Content

Chapter (6)

Resonance

6.1 Introduction

6.2 Frequency Effects on AC circuits

6.3 Series Resonance

6.4 Quality Factor, Q

6.5 Impedance of a Series Resonant Circuit

6.6 Power, Bandwidth, and Selectivity of a Series Resonant

Circuit

6.7 Series-to-Parallel RL and RC Conversion

6.8 Parallel Resonance

Series-to-Parallel RL and RC Conversion

Series-to-Parallel RL and RC Conversion

Series-to-Parallel RL and RC Conversion

Prove ????

Series-to-Parallel RL and RC Conversion

The previous two Equations may be simplified by using the quality

factor of the coil.

𝑄 =𝑋

𝑅

Series-to-Parallel RL and RC Conversion

𝑄 =𝑋𝐿𝑆𝑅𝑆

𝑄 =𝑋𝐿𝑆𝑅𝑆

=𝑅𝐿𝑝𝑋𝐿𝑝

Example 6–7

Refer to the circuit of the following figure, find the Q of the coil at ω =

1000 rad/s and convert the series RL network into its equivalent parallelnetwork. Repeat the above steps for ω = 10 krad/s.

Series-to-Parallel RL and RC Conversion

Example 6–7

Refer to the circuit of the following figure, find the Q of the coil at ω =

1000 rad/s and convert the series RL network into its equivalent parallelnetwork. Repeat the above steps for ω = 10 krad/s.

Series-to-Parallel RL and RC Conversion

Example 6–8

Series-to-Parallel RL and RC Conversion

Find the Q of each of the networks of the following figure and

determine the series equivalent for each

Example 6–8

Series-to-Parallel RL and RC Conversion

Find the Q of each of the networks of the following figure and

determine the series equivalent for each

Parallel Resonance

Parallel Resonance

The parallel resonant circuit is best analyzed using a constant-current source,

unlike the series resonant circuit which used a constant-voltage source.

Parallel Resonance

Consider the LC “a resonant circuit, tank circuit, or tuned circuit” circuit

shown in Figure 6.19. The tank circuit consists of a capacitor in parallel with an

inductor. Due to its high Q and frequency response, the tank circuit is used

extensively in communications equipment such as AM, FM, and television

transmitters and receivers.

At resonance, the capacitive and inductive reactances of the circuit are equal.

As we have observed previously, placing equal inductive and capacitive

reactances in parallel effectively results in an open circuit at the given

frequency. The input impedance of this network at resonance is therefore purelyresisitive and given as ZT = RP.

Parallel Resonance

Parallel Resonance

Notice that if R2coil << L/C, then the term under the radical is approximately equal to 1.

Consequently, if L/C ≥ 100Rcoil, the parallel resonant frequency may be simplified asc

Parallel Resonance

(j1)║(j-9) = +j0.9 (j9)║(j-1) = −j0.9

Parallel Resonance

Notice that the impedance of the entire circuit is maximum at resonance and minimum

at the boundary conditions (ω = 0 rad/s and ω → ∞). This result is exactly opposite to

that observed in series resonant circuits which have mini- mum impedance at resonance.

We also see that for parallel circuits, the impedance will appear inductive for

frequencies less than the resonant frequency, ωP. Inversely, the impedance is capacitive

for frequencies greater than ωP.

(j1)║(j-9) = +j0.9 (j9)║(j-1) = −j0.9

Parallel Resonance

The Q of the parallel circuit is determined from the definition as;

Parallel Resonance

For a parallel RLC resonant circuit, the currents in the various elements are found

from Ohm’s law as follows:

Very large value of

IC and IL

Parallel Resonance

In a manner similar to that used in determining the bandwidth of a series resonant

circuit, it may be shown that the half-power frequencies of a parallel resonant circuit

are;

Example 6–11Parallel Resonance

Refer to the networks of the following figure;

a. Determine the resonant frequencies, ωr (rad/s) and fr(Hz) of the tank circuit.

b. Find the Q of the circuit at resonance.

c. Calculate the voltage across the circuit at resonance.

d. Solve for currents through the inductor and the resistor at resonance.

e. Determine the bandwidth of the circuit in both radians per second and hertz.

f. Sketch the voltage response of the circuit, showing the voltage at the halfpower

frequencies.

g. Sketch the selectivity curve of the circuit showing P(watts) versus ω (rad/s).

Example 6–11Parallel Resonance

Refer to the networks of the following figure;

a. Determine the resonant frequencies, ωr (rad/s) and fr(Hz) of the tank circuit.

b. Find the Q of the circuit at resonance.

c. Calculate the voltage across the circuit at resonance.

Example 6–11Parallel Resonance

Refer to the networks of the following figure;

d. Solve for currents through the inductor and the resistor at resonance.

e. Determine the bandwidth of the circuit in both radians per second and hertz.

Example 6–11Parallel Resonance

Refer to the networks of the following figure;

f. Sketch the voltage response of the circuit, showing the voltage at the halfpower frequencies.

Since the Q of the circuit is less than 10,

the half-power frequencies are calculated

Example 6–11Parallel Resonance

Refer to the networks of the following figure;

g. Sketch the selectivity curve of the circuit showing P(watts) versus ω (rad/s).

g. The power dissipated by the circuit

at resonance is

Example 6–12Parallel Resonance

Refer to the networks of the following figure;

a. Calculate the resonant frequency, ωr, of the tank circuit.

b. Find the Q of the coil at resonance.

c. Sketch the equivalent parallel circuit.

d. Determine the Q of the entire circuit at resonance.

e. Solve for the voltage across the capacitor at resonance.

f. Find the bandwidth of the circuit in radians per second.

g. Sketch the voltage response of the circuit showing the voltage at the halfpower frequencies

Example 6–12Parallel Resonance

Refer to the networks of the following figure;

a. Calculate the resonant frequency, ωr,

of the tank circuit.

b. Find the Q of the coil at resonance.

c. Sketch the equivalent parallel circuit.

Example 6–12Parallel Resonance

Refer to the networks of the following figure;

d. Determine the Q of the entire circuit at resonance.

e. Solve for the voltage across the capacitor at resonance.

f. Find the bandwidth of the circuit in radians per second.

Example 6–12Parallel Resonance

Refer to the networks of the following figure;

g. Sketch the voltage response of the circuit showing the voltage at the halfpower frequencies

g. The voltage response curve is shown in the

following Figure. Since Q ≥ 10, the half-power

frequencies will occur at the following angular

frequencies:

Week Required1st 2nd 3rd

Chapter (1)

Methods of AC Analysis

4th Chapter (2)

Graphical Solution of DC Circuits Contains Nonlinear

Elements5th Chapter (3)

Exam-1

Circle Diagrams6th 7th

Chapter (4)

Transient Analysis of Basic Circuits

8th 9th Chapter (5)

Mid Term

Harmonics10th 11th

Chapter (6)

Resonance12th 13th

Chapter (7)

Passive Filters